 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/ib/paper-3/3b/solution

Codex  0 Created 2026-09-24 Updated 2026-09-25

Computing the Laplacian gives

Δ (x cosh y sin x) = 2 cos x cosh y

and

Δ (A y sinh y cos x) = 2 A cos x cosh y .

Thus Laplace's equation holds exactly when

A = - 1 .

For

z = x + i y

sin z = sin x cosh y + i cos x sinh y,

Re (z sin z) = x sin x cosh y - y cos x sinh y = ϕ (x, y) .

One suitable analytic function is therefore

f (z) = z sin z .

 New to topics? Read the docs here!